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      生成函数的思想怎样应用于组合恒等式？
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        <p>生成函数在证明许多组合恒等式的过程中都十分有用哦 ~<strong>不过，我不是专门做组合的，所以这里我就抛砖迎玉讲个简单的吧 ~</strong></p>
<p><strong>我们以 Chu-Vandermonde 恒等式为例。</strong>二项式展开（Binomial theorem, Binomial expansion）告诉我们</p>
<p><img src="//www.zhihu.com/equation?tex=%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha%7D%3D%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Calpha%5Cright%29_m%7D%7Bm%21%7Dx%5Em" alt="\left(1-x\right)^{-\alpha}=\sum_{m=0}^{\infty}\frac{\left(\alpha\right)_m}{m!}x^m"></p>
<p><img src="//www.zhihu.com/equation?tex=%5Cleft%281-x%5Cright%29%5E%7B-%5Cbeta%7D%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Cbeta%5Cright%29_n%7D%7Bn%21%7Dx%5En" alt="\left(1-x\right)^{-\beta}=\sum_{n=0}^{\infty}\frac{\left(\beta\right)_n}{n!}x^n"></p>
<p>上面等式的<strong>左边</strong>就相当于右边 <img src="//www.zhihu.com/equation?tex=x%5Em" alt="x^m"> 或 <img src="//www.zhihu.com/equation?tex=x%5En" alt="x^n"> 前面<strong>序列</strong>的<strong>生成函数</strong>（generating function）。把他俩乘在一起，就得到了</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Calpha%5Cright%29_m%7D%7Bm%21%7D%5Cfrac%7B%5Cleft%28%5Cbeta%5Cright%29_n%7D%7Bn%21%7Dx%5E%7Bm%2Bn%7D+%3D%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha%7D%5Cleft%281-x%5Cright%29%5E%7B-%5Cbeta%7D%3D%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha-%5Cbeta%7D" alt="\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\left(\alpha\right)_m}{m!}\frac{\left(\beta\right)_n}{n!}x^{m+n} =\left(1-x\right)^{-\alpha}\left(1-x\right)^{-\beta}=\left(1-x\right)^{-\alpha-\beta}"></p>
<p>然后，利用对 <img src="//www.zhihu.com/equation?tex=%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha-%5Cbeta%7D" alt="\left(1-x\right)^{-\alpha-\beta}"> 使用二项式定理进行展开，便得到了</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Calpha%5Cright%29_m%7D%7Bm%21%7D%5Cfrac%7B%5Cleft%28%5Cbeta%5Cright%29_n%7D%7Bn%21%7Dx%5E%7Bm%2Bn%7D+%3D%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Calpha%2B%5Cbeta%5Cright%29_m%7D%7Bm%21%7Dx%5E%7Bm%7D" alt="\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{\left(\alpha\right)_m}{m!}\frac{\left(\beta\right)_n}{n!}x^{m+n} =\sum_{m=0}^{\infty}\frac{\left(\alpha+\beta\right)_m}{m!}x^{m}"></p>
<p>左边的二重级数可以进行重排，从而得到</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Cleft%28%5Csum_%7Bn%3D0%7D%5E%7Bm%7D%5Cfrac%7Bm%21%7D%7B%5Cleft%28m-n%5Cright%29%21n%21%7D+%5Cleft%28%5Calpha%5Cright%29_%7Bm-n%7D%5Cleft%28%5Cbeta%5Cright%29_n%5Cright%29%5Cfrac%7Bx%5E%7Bm%7D%7D%7Bm%21%7D+%3D%5Csum_%7Bm%3D0%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Cleft%28%5Calpha%2B%5Cbeta%5Cright%29_m%7D%7Bm%21%7Dx%5E%7Bm%7D" alt="\sum_{m=0}^{\infty}\left(\sum_{n=0}^{m}\frac{m!}{\left(m-n\right)!n!} \left(\alpha\right)_{m-n}\left(\beta\right)_n\right)\frac{x^{m}}{m!} =\sum_{m=0}^{\infty}\frac{\left(\alpha+\beta\right)_m}{m!}x^{m}"></p>
<p>比较系数以后，我们便得到了：</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bn%3D0%7D%5E%7Bm%7D%5Cfrac%7Bm%21%7D%7B%5Cleft%28m-n%5Cright%29%21n%21%7D+%5Cleft%28%5Calpha%5Cright%29_%7Bm-n%7D%5Cleft%28%5Cbeta%5Cright%29_n+%3D%5Cleft%28%5Calpha%2B%5Cbeta%5Cright%29_m" alt="\sum_{n=0}^{m}\frac{m!}{\left(m-n\right)!n!} \left(\alpha\right)_{m-n}\left(\beta\right)_n =\left(\alpha+\beta\right)_m"></p>
<p>用组合数来表示的话就有</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bn%3D0%7D%5E%7Bm%7D%5Cbinom%7Bm%7D%7Bn%7D+%5Cleft%28%5Calpha%5Cright%29_%7Bm-n%7D%5Cleft%28%5Cbeta%5Cright%29_n+%3D%5Cleft%28%5Calpha%2B%5Cbeta%5Cright%29_m" alt="\sum_{n=0}^{m}\binom{m}{n} \left(\alpha\right)_{m-n}\left(\beta\right)_n =\left(\alpha+\beta\right)_m"></p>
<p>或者，利用 <img src="//www.zhihu.com/equation?tex=%5Cbinom%7B-%5Clambda%7D%7Bn%7D%3D%5Cleft%28-1%5Cright%29%5En%5Cfrac%7B%5Cleft%28%5Clambda%5Cright%29_n%7D%7Bn%21%7D" alt="\binom{-\lambda}{n}=\left(-1\right)^n\frac{\left(\lambda\right)_n}{n!}"> 将其写作</p>
<p><img src="//www.zhihu.com/equation?tex=%5Csum_%7Bn%3D0%7D%5Em%5Cbinom%7B%5Calpha%7D%7Bm-n%7D%5Cbinom%7B%5Cbeta%7D%7Bn%7D%3D%5Cbinom%7B%5Calpha%2B%5Cbeta%7D%7Bm%7D" alt="\sum_{n=0}^m\binom{\alpha}{m-n}\binom{\beta}{n}=\binom{\alpha+\beta}{m}"></p>
<p>这里主要的<strong>关键步骤</strong>是</p>
<p><img src="//www.zhihu.com/equation?tex=%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha%7D%5Cleft%281-x%5Cright%29%5E%7B-%5Cbeta%7D%3D%5Cleft%281-x%5Cright%29%5E%7B-%5Calpha-%5Cbeta%7D" alt="\left(1-x\right)^{-\alpha}\left(1-x\right)^{-\beta}=\left(1-x\right)^{-\alpha-\beta}"></p>
<p>也就是说，<strong>等式的两端可以通过两种不同的方式进行计算（这是常用的技巧）</strong>。其他非常多组合数的等式都是用这种方法得到的。除此之外，许多有关多项式序列的等式也是通过这种方法得到的，例如 Bernoulli 多项式，Euler 多项式等等。</p>
<p>来源：知乎 <a href="http://www.zhihu.com" target="_blank" rel="noopener">www.zhihu.com</a></p>
<p>作者：<a href="http://www.zhihu.com/people/luo-min-jie?utm_campaign=rss&amp;utm_medium=rss&amp;utm_source=rss&amp;utm_content=author" target="_blank" rel="noopener">罗旻杰</a></p>
<p>【知乎日报】千万用户的选择，做朋友圈里的新鲜事分享大牛。<br>        <a href="http://daily.zhihu.com?utm_source=rssyanwenzi&amp;utm_campaign=tuijian&amp;utm_medium=rssnormal" target="_blank" rel="noopener">点击下载</a></p>
<p>此问题还有 <a href="http://www.zhihu.com/question/271885602/answer/364344441?utm_campaign=rss&amp;utm_medium=rss&amp;utm_source=rss&amp;utm_content=title" target="_blank" rel="noopener">2 个回答，查看全部。</a></p>

      
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